Stat 432: Homework 3
1. The number of telephone calls coming into a switchboard during a 10 minute period
is a Poisson random variable with an average rate of 4.
a. What is the probability that exactly 1 call comes in during a 10 minute period?
b. Exactly 1 call comes in during a 10 minute period. What is the probability that
the call comes in during the first 2 minutes?
c. Exactly 1 call comes in during a 10 minute period. What is the expected time for
that call?
2. The proportion of time per hour that all checkout counters in a supermarket are busy
is a random variable, X, with density function given below.
f ( x )  cx(1  x ) 2
0
0 x 1
otherwise
a. Find c.
b. Give the distribution function for X, e.g F ( x ) .
c. What is the probability that more than 60% of the time all checkout counters are
busy?
d. What are E ( X ) and Var ( X ) ?
e. Verify that for this random variable E ( X )   1  F ( x )dx
1
0
3. One-hour carbon monoxide concentrations in air samples in a small city are modeled
with an exponential random variable with mean 2 parts per million.
a. What is the probability that the one-hour carbon monoxide concentration exceeds
5 parts per million?
b. What is the probability that the one-hour carbon monoxide concentration is within
one standard deviation of the mean?
4. Let X be an exponentially distributed random variable with parameter  . Define a
new random variable
Y  k if k  1  X  k for k  1, 2, 3, 4, 
a. Find Pr(Y  k ) .
b. The random variable Y is a common discrete random variable. What discrete
random variable is it? Be specific.